Clinical trials of fatal diseases often focus on one or more non-fatal events, in addition to survival, both to characterize morbidity and to improve survival estimates. Three statistical complications are that the time to each non-fatal event and subsequent residual survival may be either positively or negatively associated, the times to death with or without an antecedent event often have very different distributions, and death may censor some of the non-fatal event times. Consequently, the overall survival time distribution is a mixture of the distributions corresponding to the possible antecedent non-fatal events. These conditions violate the usual assumptions underlying many statistical methods for analysing multivariate time-to-event data. In this paper, we consider a general parametric model for multiple non-fatal competing risks and death. The model accounts for positive or negative association between the time of each non-fatal event and subsequent survival while accommodating covariates and the usual administrative censoring. Each event time distribution is specified marginally by a three-parameter generalized odds rate model, and the time of each non-fatal event and subsequent residual survival are combined under a bivariate generalized von Morgenstern distribution. The approach is illustrated by application to two data sets from clinical trials in colon cancer and acute leukaemia.